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A note on a theorem of Raubenheimer and Rode. (English) Zbl 1055.46032

H. Raubenheimer and S. Rode [Indag. Math., New Ser. 7, No. 4, 489–502 (1996; Zbl 0887.46026)] proved that if \(A\) is a unital complex Banach algebra which is ordered by an algebra wedge \(W\) and the spectral radius is increasing on \(W\), then \(r(a)\in \sigma(a)\). Here the authors obtain a converse result, namely, \(r(a)\in\sigma(a)\) implies the existence of a suitable wedge \(W\).

MSC:

46H05 General theory of topological algebras

Citations:

Zbl 0887.46026
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References:

[1] Sterling K. Berberian, Lectures in functional analysis and operator theory, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, No. 15. · Zbl 0296.46002
[2] H. Raubenheimer and S. Rode, Cones in Banach algebras, Indag. Math. (N.S.) 7 (1996), no. 4, 489 – 502. · Zbl 0887.46026 · doi:10.1016/S0019-3577(97)89135-5
[3] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
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